Search Results (24)

The transition of the mining industry towards Industry 5.0 demands predictive models capable of strictly adhering to physical laws and modeling complex, non-Euclidean geometries—capabilities often lacking in standard graph neural networks. This systematic review, conducted under the PRISMA 2020 protocol, analyzes the emergence of “Era 5” architectures by synthesizing 96 high-impact studies from 2019 to 2026, focusing on Clifford (geometric algebra) GNNs, simplicial and cell complex neural networks, symplectic/Hamiltonian GNNs, and generative flow networks (GFlowNets). The analysis demonstrates that Clifford architectures provide superior rotational equivariance for robotic control; Simplicial networks capture high-order topological interactions critical for geomechanics; Symplectic GNNs ensure energy conservation for stable long-term simulation of structural dynamics; and GFlowNets offer a novel paradigm for generative mine planning. We conclude that shifting from data-driven approximations to these mathematically rigorous, structure-preserving architectures is fundamental for developing reliable, physics-informed digital twins that optimize structural integrity and operational efficiency in complex industrial environments. Full article

This article examines how ethnographic methodology and literary theory can advance research engines and artificial intelligence systems beyond the reductive computational approaches that dominate contemporary AI development. Drawing on recent Stanford research revealing fundamental gaps in large language models’ ability to distinguish factual knowledge from belief, I argue that contemporary AI systems enact what I term “abducted semantics”—appropriating the inferential logic of human meaning-making while systematically attenuating the culturally embedded, phenomenologically grounded capacities that generate authentic understanding. Through close analysis of Clifford Geertz’s thick description, Charles Sanders Peirce’s triadic semiotics, and canonical literary works—Miguel de Cervantes’ Don Quixote and Gabriel García Márquez’s One Hundred Years of Solitude—I demonstrate that human understanding operates through complex semiotic processes irreducible to pattern-matching and statistical prediction. The article proposes concrete interventions to transform research engines from tools of semantic extraction into technologies that preserve and enhance interpretive richness, arguing that ethnographic and literary methodologies offer essential correctives to the epistemological impoverishment inherent in current AI architectures. Full article

Scators form a linear space equipped with a specific non-distributive product. In the elliptic case they can be interpreted as a kind of hypercomplex number. The requirement that the scator partial derivatives are direction-independent leads to a generalization of the Cauchy–Riemann equation and to scator holomorphic functions. In this paper we find a complete set of $C^2$-solutions to the generalized Cauchy–Riemann system in the $(1+n)$-dimensional elliptic scator space. For any $n⩾2$ this set consists of three classes: components exponential functions (already known), a new class of affine linear functions, and some exceptional solutions parameterized by arbitrary functions of one variable. We show, however, that the last class of solutions is not scator holomorphic and the generalized Cauchy–Riemann system should be supplemented with additional constraints to avoid such spurious solutions. The obtained family of scator holomorphic functions, although relatively narrow, is greater than that of analogous functions in quaternionic or Clifford analysis. Full article

Motivation. Analytic function theory on commutative complex extensions calls for an operator–theoretic calculus that simultaneously sees the algebra-induced coupling among components and supports boundary-to-interior mechanisms. Gap. While Dirac-type frameworks are classical in several complex variables and Clifford analysis, a coherent calculus aligning structural CR systems, a canonical first derivative, and a Cauchy-type boundary identity on the commutative model $\mathbb{C}\left(\mathbb{C}\right)\cong {\mathbb{C}}^4$ has not been systematically developed. Purpose and Aims. This paper develops such a calculus for $\mathcal{O}$-regular mappings on $\mathbb{C}\left(\mathbb{C}\right)$ and establishes three pillars of the theory. Main Results. (i) A fully coupled Cauchy–Riemann system characterizing $\mathcal{O}$-regularity; (ii) identification of a canonical first derivative $g^{\prime }\left(z\right)={\partial }_{x_0}g\left(z\right)$; and (iii) a Stokes-driven boundary annihilation law ${\int }_{\partial \mathrm{\Omega }}\tau g=0$ for a canonical 7-form $\tau$. On (pseudo)convex domains, $\overline{\partial }$-methods yield solvability under natural compatibility and regularity assumptions. Stability (under algebra-preserving maps), Liouville-type, and removability results are also obtained, and function spaces suited to this algebra are outlined. Significance. The results show that a large portion of the classical holomorphic toolkit survives, in algebra-aware form, on $\mathbb{C}\left(\mathbb{C}\right)$. Full article

The simulation of multidimensional wave propagation with variable material parameters is a computationally intensive task, with applications from seismology to electromagnetics. While quantum computers offer a promising path forward, their algorithms are often analyzed in the abstract oracle model, which can mask the high gate-level complexity of implementing those oracles. We present a framework for constructing a quantum algorithm for the multidimensional wave equation with a variable speed profile. The core of our method is a decomposition of the system Hamiltonian into sets of mutually commuting Pauli strings, paired with a dedicated diagonalization procedure that uses Clifford gates to minimize simulation cost. Within this framework, we derive explicit bounds on the number of quantum gates required for Trotter–Suzuki-based simulation. Our analysis reveals significant computational savings for structured block-model speed profiles compared to general cases. Numerical experiments in three dimensions confirm the practical viability and performance of our approach. Beyond providing a concrete, gate-level algorithm for an important class of wave problems, the techniques introduced here for Hamiltonian decomposition and diagonalization enrich the general toolbox of quantum simulation. Full article

We develop a rigorous algebraic–analytic framework for multidual complex numbers ${\mathbb{DC}}_n$ within the setting of Clifford analysis and establish a comprehensive theory of hyperholomorphic multidual complex-valued functions. Our main contributions are (i) a fully coupled multidual Cauchy–Riemann system derived from the Dirac operator, yielding precise differentiability criteria; (ii) generalized conjugation laws and the associated norms that clarify metric and geometric structure; and (iii) explicit operator and kernel constructions—including generalized Cauchy kernels and Borel–Pompeiu-type formulas—that produce new representation theorems and regularity results. We further provide matrix–exponential and functional calculus representations tailored to ${\mathbb{DC}}_n$, which unify algebraic and analytic viewpoints and facilitate computation. The theory is illustrated through a portfolio of examples (polynomials, rational maps on invertible sets, exponentials, and compositions) and a solvable multidual boundary value problem. Connections to applications are made explicit via higher-order automatic differentiation (using nilpotent infinitesimals) and links to kinematics and screw theory, highlighting how multidual analysis expands classical holomorphic paradigms to richer, nilpotent-augmented coordinate systems. Our results refine and extend prior work on dual/multidual numbers and situate multidual hyperholomorphicity within modern Clifford analysis. We close with a concise summary of notation and a set of concrete open problems to guide further development. Full article

This work addresses a significant gap in the existing literature by developing integral representation formulas for extended quaternion-valued functions within the framework of Clifford analysis. While classical Cauchy-type and Borel–Pompeiu formulas are well established for complex and standard quaternionic settings, there is a lack of analogous tools for functions taking values in extended quaternion algebras such as split quaternions and biquaternions. The motivation is to extend the analytical power of Clifford analysis to these broader algebraic structures, enabling the study of more complex hypercomplex systems. The objectives are as follows: (i) to construct new Cauchy-type integral formulas adapted to extended quaternionic function spaces; (ii) to identify explicit kernel functions compatible with Clifford-algebra-valued integrands; and (iii) to demonstrate the application of these formulas to boundary value problems and potential theory. The proposed framework unifies quaternionic function theory and Clifford analysis, offering a robust analytic foundation for tackling higher-dimensional and anisotropic partial differential equations. The results not only enhance theoretical understanding but also open avenues for practical applications in mathematical physics and engineering. Full article

Parkinson’s disease (PD) is a chronic neurodegenerative disorder that progressively impairs motor functions. Clinical assessments have traditionally relied on rating scales such as the Movement Disorder Society Unified Parkinson Disease Rating Scale (MDS-UPDRS); however, these evaluations are susceptible to rater-dependent variability and may miss subtle motor changes. This study explored objective and quantitative methods for assessing motor function in PD patients using the Quantum Metaglove, a sensory glove produced by MANUS^®, which was used to record finger movements during three tasks: finger tapping, hand gripping, and pronation–supination. Classic and geometric motor features (the latter based on Clifford algebra, an advanced approach for trajectory shape analysis) were extracted. The resulting data were used to train various machine learning algorithms (k-NN, SVM, and Naive Bayes) to distinguish healthy subjects from PD patients. The integration of traditional kinematic and geometric approaches improves objective hand movement analysis, providing new diagnostic opportunities. In particular, geometric trajectory analysis provides more interpretable information than conventional signal processing methods. This study highlights the value of wearable technologies and Clifford algebra-based algorithms as tools that can complement clinical assessment. They are capable of reducing inter-rater variability and enabling more continuous and precise monitoring of hand motor movements in patients with PD. Full article

This paper develops a theoretical framework for the computation of higher-order derivatives based on the algebra of hyper-dual numbers. Extending the classical dual number system, hyper-dual numbers provide a natural and rigorous mechanism for encoding and propagating derivative information through function composition and arithmetic operations. We formalize the underlying algebraic structure, define generalized hyper-dual extensions of scalar functions via multidimensional Taylor expansions, and establish consistency with standard differential calculus. The proposed approach enables exact computation of partial derivatives and mixed higher-order derivatives without resorting to symbolic manipulation or approximation schemes. We further investigate the algebraic properties and closure under differentiable operations, illustrating the method’s potential for unifying aspects of automatic differentiation with multivariable calculus. This study contributes to the theoretical foundation of algorithmic differentiation and highlights hyper-dual numbers as a precise and elegant tool in computational differential analysis. Full article

In the framework of harmonic and Clifford analysis, specific distributions in Euclidean space of arbitrary dimension, which are of particular importance for theoretical physics, are once more thoroughly studied. Indeed, actions involving spherical coordinates, such as the radial derivative and multiplication and division by the radial distance, only make sense when considering so-called signumdistributions, that is, bounded linear functionals on a space of test functions showing a singularity at the origin. Introducing these signumdistributions, the actions of a whole series of scalar and vectorial differential operators on the distributions under consideration, lead to a number of surprising results, as illustrated by some examples in three-dimensional mathematical physics. Full article

One of the main aims of Clifford analysis is to study the growth properties of regular functions. Biregular functions are a well-known generalization of regular functions. In this paper, the growth orders and types of biregular functions are studied. First, generalized growth orders and types of biregular functions are defined in the context of Clifford analysis. Then, using the methods of Wiman and Valiron, generalized Lindelöf–Pringsheim theorems are proved, which show the relationship between growth orders, growth types, and Taylor series. These connections allow us to calculate the growth order and determine the type of biregular functions. Full article

Utilizing Multi-Criteria Decision Analysis (MCDA) methods based on environmental, social, and governance (ESG) factors to rank countries according to these criteria aims to evaluate and prioritize countries based on their performance in environmental, social, and governance aspects. The contemporary world is influenced by a multitude of factors, which consequently impact our lives. Various models are devised to assess company performance, with the intention of enhancing quality of life. An exemplary case is the ESG framework, encompassing environmental, social, and governmental dimensions. Implementing this framework is intricate, and many nations are keen on understanding their global ranking and avenues for enhancement. Different statistical and mathematical methods have been employed to represent these rankings. This research endeavors to examine both types of methods to ascertain the one yielding the optimal outcome. The ESG model comprises eleven factors, each contributing to its efficacy. We employ the Performance Contribution Analysis (PCA), Clifford algebra method, and entropy weight technique to rank these factors, aiming to identify the most influential factor in countries’ ESG-based rankings. Based on prioritization results, political stability (PSAV) and the voice of accountability (VA) emerge as pivotal elements. In light of the ESG model and MCDA methods, the following countries exhibit significant societal impact: Sweden, Finland, New Zealand, Luxembourg, Switzerland, Denmark, India, Norway, Canada, Germany, Austria, and Australia. This research contributes in two distinct dimensions, considering the global context and MCDA methods employed. Undoubtedly, a research gap is identified, necessitating the development of a novel model for the comparative evaluation of countries in relation to prior studies. Full article

The perturbed Dirac operators yield a factorization for the well-known Helmholtz equation. In this paper, using the fundamental solution for the perturbed Dirac operator, we define Cauchy-type integral operators (singular integral operators with a Cauchy kernel). With the help of these operators, we investigate generalized Riemann and Dirichlet problems for the perturbed Dirac equation which is a higher-dimensional generalization of a Vekua-type equation. Furthermore, applying the generalized Cauchy-type integral operator ${\tilde{F}}_{\lambda },$ we construct the Mann iterative sequence and prove that the iterative sequence strongly converges to the fixed point of operator ${\tilde{F}}_{\lambda }.$ Full article

The aim of this work is to study the method of the normalized systems of functions. The normalized systems of functions with respect to the Dirac operator in the umbral Clifford analysis are constructed. Furthermore, the solutions of umbral Dirac-type equations are investigated by the normalized systems. Full article

The academic discussion of charisma takes two major voices as the point of departure: Max Weber and St. Paul. In both areas, sociology and religion, charisma is seen as a quality of persons. My argument is that entire cultures can be suffused by this force, and that social life, education, and modes of expression can be bearers and transmitters of charismatic force. I approach the argument conceptually, drawing on a remarkable passage in Goethe’s autobiography Dichtung und Wahrheit. What Goethe calls “the demonic” is charisma conceived as a force that can penetrate, unpredictably, either natural phenomena or persons. To these I add institution, cultures, and structures of government. The charisma of larger structures, like personal charisma, has a life-cycle, charisma in its cultural structuring being as unstable as in its personal embodiment. The idea opens cultural transformations to analysis. Clifford Geertz has provided a model. The sea-changes that transformed western European culture from the twelfth to the thirteenth century show us the end of a life-cycle of charismatic culture, and the transition to intellectual or textual culture. Charisma moved out of the realm of the lived and expressed social forms and into art and artifice, rationalizing philosophy, theology, liturgy and other forms of Christian discourse (sermons). Three voices from the later thirteenth century observe this development closely—the loss of charisma as a political–social–cultural force—and lament the loss. Full article